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Full text of "Men Of Mathematics"

MEN OF MATHEMATICS

complex integers (numbers of the form a  bi , where a, 6 are
rational integers and i denotes V  1) into the higher arith-
metic in order to give the law of biquadratic reciprocity its
simplest expression. Dirichlet and other followers of Gauss then
discussed quadratic forms in which- the rational integers
appearing as variables and coefficients are replaced by Gaussian
complex integers. Herrnite passed to the general case of this
situation and investigated the representation of integers in
what are to-day called Hermitian forms. An example of such a
form (for the special case of two complex variables as^ x* and
their 'conjugates* x^ x instead of n variables) is

+ fl22

in 'which the bar over a letter denoting a complex number indi-
cates the conjugate of that number; namely, if a? -f- iy is the
complex number, its 'conjugate' is x  iy; and the coefficients
<*ii i2t asi> fl22 are such that a% = ayt, for (ij) = (1,1), (1,2),
(2,1), (2,2), so that a12 and azl are conjugates, and each of <zu,
aZ2 is its own conjugate (so that an, a22 are real numbers). It is
easily seen that the entire form is real (free of i) if all products
are multiplied out, but it is most 'naturally' discussed in the
shape given.
\Vhen Hermite invented such forms he was interested in
finding what numbers are represented by the forms. Over
seventy years later it was found that the algebra of Hermitian
forms is indispensable in mathematical physics, particularly in
the modern quantum theory. Hermite had no idea that his pure
mathematics would prove valuable in science long after his
death. - indeed, like Archimedes, he never seemed to care much
for the scientific applications of mathematics. But the fact that
Hermite's work has given physics a useful tool is perhaps
another argument favouring the side that believes mathemati-
cians best justify their abstract existence when left to their own
inscrutable devices.
Leaving aside Hennite's splendid discoveries in the theory of
algebraic invariants as too technical for discussion here, we
shall pass on in a moment to two of his most spectacular achieve-
ments in other fields. The high esteem in which Hennite's
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